Displacement interaction model, \(v_{p,p^{\prime }}^B=0\)  and
\(\zeta _{p,p^{\prime }}^B=0\) . Given the remaining linear displacement-polarization
interaction, an analytic form for the bosonic IF can be written, the
so-called “Feynman-Vernon" influence functional (FV IF) [1]}.
In its time-discrete form, the bosonic IF is given by an exponent
with pairwise interactions along the path [2]}, 
\(I_B(s_0^{\pm },..., s_{N}^{\pm }) =\exp \left[ -\sum _{k=0}^{N}\sum _{k^{\prime }=0}^k(s_k^+-s_k^-)(\eta _{k,k^{\prime }}s_{k^{\prime }}^+-\eta _{k,k^{\prime }}^*s_{k^{\prime }}^-)\right].\) 
